Here is the full list of all 14 Ring Run Circus trophies - there are 11 bronze, 2 silver and 1 gold.
Get on stage.
Swap characters 50 times
The fewest ring movements to solve "A Triple Puzzle"
Ride one turn on the carousel
Complete "A Bearded Ball" without getting burned
Three on the floor at once in "An Encirclement"
Complete "The Jump" with no stars
Complete "A Lucky Bounce" without bouncing
Get 50 speed boosts in a row
Hit two with a single punch in "The Lions"
Avoid The Magician in "The Flight On Fire"
The fewest character-swaps to solve "The Swap"
Hit someone with the ball
Get 10 speed boosts in a row